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Why is it so important that the sun is directly overhead at noon?
First of all, the sun is rarely directly overhead at noon anymore, even during standard time, because we use the same time for all the time in the time zone, so the sun is only directly overhead at noon if you are in the center of the time zone. (And even then, the sun is not always overhead at noon all the time- see https://en.m.wikipedia.org/wiki/Equation_of_time )
Have you ever noticed that the sun is never directly overhead at noon, even when you think it should be? I doubt it.
Second of all, Noon is no longer mid-day anyway. The vast majority of people are awake for fewer hours before noon than after noon.
Permanent DST basically means that we are now considering 1PM to be mid-day, which is more compatible with most people's schedules.
I want to point out that solar noon is the term to describe when the sun reaches its highest point in the sky.
Because that's what noon means. There should at least be a chance.
Noon hasn't meant that the sun is directly overhead since November 18, 1883.
https://guides.loc.gov/this-month-in-business-history/november/day-of-two-noons
And technically, Ptolemy knew in the 2nd century that mean solar time and apparent solar time were different, so you could argue we have known it for nearly 1900 years.
What is important isn't that the sun is directly overhead, just that it is "close" to overhead. Wherever you live you plan your day around the sun. Sure we have lights, and if you never go outside they are fine - but most people would agree that is a terrible life: once you accept you will go outside once in a while you want the times you go outside to be when the sun is up if possible (except astronomers who what nobody else to have those artificial lights on when it should be dark)
In the Northern Hemisphere, one can always expect the sun to be somewhat south of overhead at noontime. The close to the equator you live, the more directly overhead it will be.
The tropics is the only area on Earth where the Sun will ever be directly overhead. The sun's average position is directly over the equator and the wobble in the Earth's rotation let's it's absolute position vary about 23.5° north or south which is what defines the tropics.